## Here's a little tutorial on how to use the `factanal` function for Factor Analysis.
## Let's make a data frame using actual factors:
set.seed(3847)
F1 <- rnorm(100)
F2 <- rnorm(100)
X1 <- F1 + F2 + rnorm(100)/5
X2 <- 4*F1 * rnorm(100)
X3 <- -6*F1 * rnorm(100)/4
X4 <- F2 + rnorm(100)
X5 <- 4*F2 - 0.5*F1 * rnorm(100)
dat <- data.frame(X1=X1, X2=X2, X3=X3, X4=X4, X5=X5)
## Let's do a factor analysis with k=2 factors. There are two main ways you can do this.
k <- 2
## Method 1: via the data.
## Just indicate the data in the first argument, and specify the number of factors
## in the `factors` argument.
## Note that loadings that are "essentially zero" are indicated as blanks.
factanal(dat, factors=k)
##
## Call:
## factanal(x = dat, factors = k)
##
## Uniquenesses:
## X1 X2 X3 X4 X5
## 0.375 0.005 0.947 0.487 0.035
##
## Loadings:
## Factor1 Factor2
## X1 0.790
## X2 0.160 0.985
## X3 0.119 -0.198
## X4 0.715
## X5 0.976 -0.110
##
## Factor1 Factor2
## SS loadings 2.128 1.024
## Proportion Var 0.426 0.205
## Cumulative Var 0.426 0.630
##
## Test of the hypothesis that 2 factors are sufficient.
## The chi square statistic is 0.93 on 1 degree of freedom.
## The p-value is 0.336
## Method 2: via the covariance matrix.
## Indicate the covariance matrix in the `covmat` argument.
## Note: this method doesn't have as many features, such as hypothesis testing,
## built into it. But the results are the same.s
factanal(factors=k, covmat=cov(dat))
##
## Call:
## factanal(factors = k, covmat = cov(dat))
##
## Uniquenesses:
## X1 X2 X3 X4 X5
## 0.375 0.005 0.947 0.487 0.035
##
## Loadings:
## Factor1 Factor2
## X1 0.790
## X2 0.160 0.985
## X3 0.119 -0.198
## X4 0.715
## X5 0.976 -0.110
##
## Factor1 Factor2
## SS loadings 2.128 1.024
## Proportion Var 0.426 0.205
## Cumulative Var 0.426 0.630
##
## The degrees of freedom for the model is 1 and the fit was 0.0097
## Another important thing to note is how to do *rotations*. There are two
## built-in rotations available via the `rotations` argument -- they are
## "none" and "varimax" (default). So the plain factor analysis results
## without rotation is:
factanal(dat, factors=k, rotation = "none")
##
## Call:
## factanal(x = dat, factors = k, rotation = "none")
##
## Uniquenesses:
## X1 X2 X3 X4 X5
## 0.375 0.005 0.947 0.487 0.035
##
## Loadings:
## Factor1 Factor2
## X1 0.784 0.101
## X2 0.997
## X3 0.151 -0.175
## X4 0.712
## X5 0.981
##
## Factor1 Factor2
## SS loadings 2.106 1.046
## Proportion Var 0.421 0.209
## Cumulative Var 0.421 0.630
##
## Test of the hypothesis that 2 factors are sufficient.
## The chi square statistic is 0.93 on 1 degree of freedom.
## The p-value is 0.336
## You can extract the loadings vector by extracting the list component
## entitled `loadings`. There's a weird print call associated with the
## object, but it's really just a matrix, as the last call indicates.
fit <- factanal(dat, factors=k, rotation = "none")
fit$loadings
##
## Loadings:
## Factor1 Factor2
## X1 0.784 0.101
## X2 0.997
## X3 0.151 -0.175
## X4 0.712
## X5 0.981
##
## Factor1 Factor2
## SS loadings 2.106 1.046
## Proportion Var 0.421 0.209
## Cumulative Var 0.421 0.630
fit$loadings[1:5, 1:2]
## Factor1 Factor2
## X1 0.783885737 0.10136519
## X2 -0.009679362 0.99744995
## X3 0.151115775 -0.17503806
## X4 0.711852604 0.08077992
## X5 0.980910146 0.05733956
)